Manipulatives and number sentences in computer aided arithmetic word problem solving

In this study we investigated the relative contribution of two main components often used in the instruction of arithmetic word problem solving to first-grade children and children with learning problems, external representation with manipulatives and formal mathematical representation with number sentences. Four computer aided treatments were developed along these dimensions. Furthermore, an assessment control group was formed.It was shown that performance improved significantly after treatment in comparison to the performance of the assessment control group. Comparison of the different training conditions showed that a number sentence treatment was the most effective training component. However, this effect was only present with the relatively high competent children and not with the relatively low performing children.     

聋生与听力正常学生构建加减文字题问题模型的比较研究

运用实验法,研究了聋生与听力正常学生解决加减文字题、构建加减文字题问题模型的差异。被试为小学3年级学生,聋生32名,听力正常学生33名。结果发现:聋生解决加减文字题和构建问题模型的成绩都显著落后于听力正常学生;构建问题模型在聋生解决加减文字题的过程中有着重要的作用。研究认为,在聋校文字题教学中要重视问题表征训练。     

中美学生数学问题解决的差异比较

“问题解决”是国际数学教育研究的一个热点,本文从中美两国学生在问题解决的过程和结果的比较研究入手,比较和研究了中美两国学生在问题解决中提出问题的能力及解决问题的策略两方面的差异。     

The consistency effect in word problem solving is effectively reduced through verbal instruction

In mathematical word problem solving, a relatively well-established finding is that more errors are made on word problems in which the relational keyword is inconsistent instead of consistent with the required arithmetic operation. This study aimed at reducing this consistency effect. Children solved a set of compare word problems before and after receiving a verbal instruction focusing on the consistency effect (or a control verbal instruction). Additionally, we explored potential transfer of the verbal instruction to word problems containing other relational keywords (e.g., larger/smaller than) than those in the verbal instruction (e.g., more/less than). Results showed a significant pretest-to posttest reduction of the consistency effect (but also an unexpected decrement on marked consistent problems) after the experimental verbal instruction but not after the control verbal instruction. No significant effects were found regarding transfer. It is concluded that our verbal instruction was useful for reducing the consistency effect, but future research should address how this benefit can be maintained without hampering performance on marked consistent problems.     

Teaching and testing mathematical problem solving by offering optional assistance

Transfer capability is usually defined as theability to apply acquired knowledge and skillsin novel situations.The experiment reported here concerned transferin mathematics education. An experimental programme was constructed, based on strengtheningthe connection of strategic and domain specificknowledge and offering hints during teaching aswell as during testing.Subjects were first graders from secondaryeducation in the Netherlands, from two schools,two classes each. Students from these fourclasses were randomly allocated either to theexperimental or the control group.The experimental computer-supported teachingprogramme was offered once a week during sixweeks; the control group received regularmathematics instruction.After controlling for the effects of thecovariates intelligence, mathematics aptitudeand anxiety, it was shown that the experimentalsubjects performed significantly better on aposttest than subjects in the control group.The results suggest that the experimentalinstruction method enhances mathematics problemsolving ability more strongly than traditionalinstruction. Upon closer examination thiseffect appears to be restricted to subjectsalready relatively high in intelligence andmathematical ability. This finding is notuncommon in intervention research, and issometimes referred to as the Matthew orthe fan-spread effect.     

聋童与普通儿童解决加减文字题和字词意识关系的比较研究

探讨考察聋童与普通儿童在解决加减文字题上是否存在差异以及这种差异与字词意识发展的关系。被试为二、三年级的普通儿童51人和三、四、五年级的聋童49人。研究结果表明:(1)聋童解决加减文字题的正确率显著地低于普通儿童,聋童和普通儿童解决合并题的正确率均好于变化题和比较题;(2)聋童和普通儿童在解决三种类型的加减文字题上均存在着年级差异;(3)聋童字词意识好的比例低于普通儿童;(4)字词意识对聋童的影响小于对普通儿童的影响。     

Tracing the development of representational flexibility and problem solving in fraction addition: a longitudinal study

The study models the development of students’ multiple-representation flexibility and the use of problem-solving strategies and representations in fraction addition. A test administered three times, with breaks of 3–4?months between successive measurements to 108 students at a transition within primary school (Grade 5–6), 132 students at a transition from primary to secondary education (Grade 6–7) and 148 students at a transition within secondary school (Grade 7–8). Multivariate analysis of variance for repeated measures and dynamic structural equation modelling were carried out in order to analyse the data. Findings suggested that students’ performance improve through measurements. Dynamic modelling provided evidence for the strong interrelation between representational flexibility and problem solving at the three measurements. The results indicated the students’ established pre-existent knowledge and the important role the initial state of the aforementioned cognitive parameters plays on their advancement. Didactical implications are discussed.     

有关数学问题的解题思想与方法

我们从中学就开始接触各类数学问题,而要解决这些数学问题,最重要的就是找出问题的精髓也就是所运用的思想与方法,并且这些思想与方法在实际应用中也非常广泛,因此,在这里我们主要介绍几种重要的解决数学问题的思想与方法。     

Middle school children's strategic behavior: Classification and relation to academic achievement and mathematical problem solving

Theories of problem solving (e.g., Verschaffelet al., 2000) hold strategic behavior centralto processing mathematical word problems. Thepresent study explores 80 sixth- andseventh-grade students' self-reported use of 14categories of strategies (Zimmerman &Martinez-Pons, 1986) and the relationship ofstrategy use to academic achievement,problem-solving behaviors, and problem-solvingsuccess. High and low achievement groupsdiffered in the number of different strategiesand categories of strategies reported but notin overall number of strategies, confidence inusing strategies, or frequency of strategy use.Students whose behaviors evidenced elaborationof the word problem's text reported moreself-evaluation; organizing and transforming;and goal setting and monitoring behavior.Implications for instructional practices thatsupport active stances toward problem solvingare discussed.     

Creativity and inspiration for problem solving in engineering education

Problem solving is a critical skill for engineering students and essential to development of creativity and innovativeness. Essential to such learning is an ease of communication and allowing students to address the issues at hand via the terminology, attitudes, humor and empathy, which is inherent to their frame of mind as novices, without the attempt to have to be the expert. Deep learning of scientific fact can be facilitated by using non-conventional tools for teaching, learning and presentation such as drama, video, posters, model making and other similar means. It may be time to break free of the PowerPoint tradition to generate successful approaches for establishing student engagement and maintaining such engagement.     

The Effect of External Representations on Compare Word Problems: Supporting Mental Model Construction

This study explored the effectiveness of external representations presented together with compare word problems, and whether such effectiveness was moderated by working memory. Participants were 49 secondary school students. Each participant solved 48 problems presented in 4 presentation types that included 2 difficulty treatments (number of steps and language consistency). Data from errors and response times indicated that adding a graph to compare word problems helped accuracy (d =.60) and response time (d =.44). This effect was larger for difficult and cognitively demanding problems. Findings suggest that external representations might help problem-solvers build a mental model of the described problem situation. In addition, working memory was not found to moderate the effectiveness of external representations.     

The contribution of external representations in pre-school mathematical problem solving

Could problem solving be the object of teaching in early education? Could appropriate teaching interventions develop to scaffold children's efforts to solve problems? These were the central questions of this article. The sample consisted of 18 children attending public pre-school in Cyprus. The problem they were asked to solve was to find all solutions of the pentomino. The children's problem solving was supported by graphically representing their solutions on squared paper. The findings show that children responded positively to the problem and were successful in finding all solutions for the specific problem. The graphical representation of the solutions and the forms of teacher–children and children–children interactions played an important role in the positive outcome of the activity.     

Fostering analogical transfer: The multiple components approach to algebra word problem solving in a chemistry context

Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated multiple components (symbolic equations, symbols, categorization, hint) in the source, or target, or both, to address the four analogical tasks. Different combinations of the components were tested in a series of four experiments. Symbolic equations (main component) fostered a mental construction of the problem in its solution mode. Categorization enabled an identification of the problem category. A hint in the target directed the learners to the source problem. The interaction between these components facilitated the mapping of the symbolic equations in the source onto the target, resulting in the superiority of the MC approach in fostering analogical transfer. Neither the main component alone nor the main component plus one sub-component was sufficient for analogical transfer. Hence for analogical transfer to occur, at least the main component (symbolic equations) and two sub-components (categorization and hint) are required. However, symbols may not have additional effects for transfer to occur.     

中学数学中常用的解题思想和解题方法

阐述了中学数学中几种常用的解题思想：方程的思想、函数思想、转化思想、分类讨论思想、数形结合思想等.同时分析了几种常用的解题方法：换元法、消元法、参数法、递推法、构造法.     

Computer technology and complex problem solving: Issues in the study of complex cognitive activity

Goals and plans organize much of complex problem solving behavior and are often inferable from action sequences. This paper addresses the strengths and limitations of inferring goals and plans from information that can be derived from computer traces of software used to solve mathematics problems. We examined mathematics problem solving activity about distance, rate, time relationships in a computer software environment designed to support understanding of functional relationships among these variables (e.g., distance =rate × time; time=distance/rate) using graphical representations of the results of simulations. Ten adolescent-aged students used the software to solve two distance, rate, time problems, and provided think-aloud protocols. To determine the inferability of understanding from the action traces, coders analyzed students' understanding from the computer traces alone (Trace-only raters) and compared these to analyses based on the traces plus the verbal protocols (Traceplus raters). Inferability of understanding from the action traces was related to level of student understanding how they used the graphing tool. When students had a good understanding of distance, rate, time relationships, it could be accurately inferred from the computer traces if they used the simulation tool in conjunction with the graphing tool. When students had a weak understanding, the verbal protocols were necessary to make accurate inferences about what was and was not understood. The computer trace also failed to capture dynamic exploration of the visual environment so students who relied on the graphing tool were inaccurately characterized by the Trace-only coders. Discussion concerns types of scaffolds that would be helpful learning environment for complex problems, the kind of information that is needed to adequately track student understanding in this content domain, and instructional models for integrating learning environments like these into classrooms.Members of the Cognition and Technology Group at Vanderbilt who have contributed to this project are (in alphabetical order) Helen Bateman, John Bransford, Thaddeus Crews, Allison Moore, Mitchell Nathan, and Stephen Owens. The research was supported, in part, by grants from the National Science Foundation (NSF-MDR-9252990) but no official endorsement of the ideas expressed herein should be inferred.     

问题解决式的缘起与理论背景

在改革高等数学教学的过程中,问题解决式教学是得到关注与认可的一种模式,然而,纵览相关的文献不难发现,介绍此教学模式的过程、方法的居多,而对这一模式兴起的原因以及理论基础进行分析的相对较少,这显然不利于此教学模式的推广.为此,本文从数学问题解决研究的兴起、教学模式的理论依据入手,介绍、分析了问题式高等数学教学模式背后的一些理论知识,期望以此推动新型数学教学模式的实施.     

Problem solving in mathematics: the significance of visualisation and related working memory

The importance of educational psychologists’ (EPs’) skills to the formulation of evidence-based educational strategies, as well as in response to learning difficulties, is demonstrated here in relation to mathematical problem solving. Initiatives to improve the link between mathematical skills from school to everyday life have drawn significantly on problem solving tasks. Through critical evaluation of research, the relevance of visualisation methods and working memory to problem solving is considered within this article. Studies suggest that differences exist in the effectiveness of particular visualisation methods, but that training can improve their utility and thereby problem solving performance. Additionally, differences between individuals and contexts can influence visualisation use, and some pupils experience specific difficulties in this area. Recommendations are made to enhance the teaching of visualisation as a strategy for problem solving, and to support those pupils with specific difficulties.     

Representation in memory and acquisition of knowledge from text and picture: Theoretical,methodological, and practical outcomes

Over the last 10 years, text comprehension research has undergone profound theoretical and methodological upheaval. Today, psychologists study language and texts because they present the trace markers of structures and cognitive operations. A text is no longer seen as a vehicle for conveying constructed meaning. Rather, text has been redefined as merely a structured sequence of stimuli, which activates both domain-related knowledge triggered by text information and linguistic knowledge. We present the theoretical views and experimental findings of the TEXTIMA group, characterized by the importance of the connectionist concepts.     

Impact of authenticity on sense making in word problem solving

The study presented in this paper seeks to investigate the impact of authenticity on the students’ disposition to make necessary real world considerations in their word problem solving. The aim is also to gather information about the extent to which different reasons for the students’ behaviors are responsible for not providing solutions that are consistent with the ‘real’ situations described in the word problems. The study includes both written solutions to word problems and interview data from 161 5th graders. The results show an impact of authenticity on both the presence of ‘real life’ considerations in the solution process and on the proportion of written solutions that were really affected by these considerations. The students’ frequent use of superficial solution strategies and their beliefs about mathematical word problem solving were found to be the main reasons for providing solutions that are inconsistent with the situations described in the word problems.     

evolving research of mathematics teacher educators: the case of non-standard issues in solving standard problems

This study describes the learning of researchers who engage in mathematics teacher education as an integral part of their practice. As teacher educators working with teachers on the subject of proportional reasoning, the authors reflected on teachers solutions to a standard problem and analyzed answers that would conventionally be considered incorrect. This exploration showed that some incorrect answers made sense, were based on problem situation analysis, and brought attention to the fact that conventional formal answers were given without much deliberation on their meaning in the situation. This insight prompted a second research phase in which teachers discussed and explained alternative solutions, and developed deeper analysis of problem situation in solutions that had been correct in the first place. The importance of making teachers aware of the nature of alternative solutions was further exhibited in a third research phase in which teachers evaluated childrens answers to the same problem. The pedagogical insight that emerged stressed the importance of making teachers aware of the tension between an almost automatic application of a mathematical model, and of analyzing problem situation during problem solving. In addition, the researchers developed better understanding of the mathematical challenge associated with the proportional reasoning problem, a stronger awareness of the role of sensitivity to their learners (the teachers), and of the role of reflection.